Abstract

In this paper we propose the addition of an artificial diffusion model to the Lagrange-Galerkin finite element method, which is dependent on both the mesh function and the numerical solution . The purpose of this additional term is to produce smooth approximations to sharp features of the solution, such as internal and boundary layers. Moreover, the added diffusion will help to stabilise the numerical scheme.

Further, we extend the a posteriori error analysis presented in Technical Report NA-95/24 to include this artificial diffusion model. Based on this a posteriori estimate, we design an adaptive algorithm to ensure global control of the error in the norm with respect to a pre-determined tolerance. The performance of this numerical algorithm is demonstrated by some numerical experiments.